I prefer to not have either foundation or an anti-foundational axiom. (Foundation generally leads to a more intuitive universe with sets sort of being like boxes but anti-foundational axioms lead to more interesting systems).

I'm also confused by cousin it's claim. I don't see how bisimulation helps one deal with Russell's paradox but I'd be interested in seeing a sketch of an attempt. As I understand it, if you try to use a notion of bisimilarity rather than extensionality and apply Russell's Paradox, you end up with essentially a set that isn't bisimilar to itself. Which is bad.